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<b>let </b><font color="Maroon">H</font> be   <a href="glib_000.html#NM5">inducedSubgraph</a> of <font color="Maroon">G</font>,<font color="Maroon">V</font>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:116_1_1_1_1"/>
(  not <font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> or <font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:116_1_1_1_1_1"/>
<font color="Maroon">V</font> is  not  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:116_1_1_1_1_1"/><b>then </b>
<font color="Maroon">H</font> <a href="glib_000.html#R5">==</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><b>hence </b><a NAME="E3:116_1_1_1_1_1"/>
<font color="Maroon">H</font> is <a href="chord.html#V6" target="_self">chordal</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="116_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:116_1_1_1_1_2"/>
<font color="Maroon">V</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6">the_Vertices_of</a> <font color="Maroon">G</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:116_1_1_1_1_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="glib_000.html#D39">GLIB_000:def 39</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="116_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">W</font> be    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">H</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:116_1_1_1_1_2_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 <b>and </b><br/><a NAME="E2:116_1_1_1_1_2_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V8">Cycle-like</a>
 ;<br/><a NAME="E3:116_1_1_1_1_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">W</font> is <a href="glib_001.html#V5">Path-like</a>
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">W</font> as    <a href="glib_001.html#M3">Walk</a> of <font color="Maroon">G</font> <b>by </b><i><a class="ref" href="glib_001.html#T168">GLIB_001:168</a></i>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="glib_001.html#NM5">Path</a> of <font color="Maroon">G</font> <b>by </b><i>, <a class="ref" href="glib_001.html#T176">GLIB_001:176</a></i>;<br/><a NAME="E6:116_1_1_1_1_2_1"/>
( <font color="Maroon">W</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a> &amp; <font color="Maroon">W</font> is <a href="glib_001.html#V5">Path-like</a> )
 <b>by </b><i>, <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E7:116_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">P</font> is <a href="glib_001.html#V1">closed</a> &amp; not <font color="Maroon">P</font> is <a href="glib_001.html#V3">trivial</a> &amp; <font color="Maroon">P</font> is <a href="glib_001.html#V5">Path-like</a> )
 <b>by </b><i><a class="ref" href="glib_001.html#T177">GLIB_001:177</a></i>;<br/><a NAME="E8:116_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">P</font> is <a href="glib_001.html#V8">Cycle-like</a>
 <b>by </b><i><a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a></i>;<br/><a NAME="E9:116_1_1_1_1_2_1"/>
<font color="Maroon">P</font> <a href="glib_001.html#K15">.length()</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 3
 <b>by </b><i>, <a class="ref" href="glib_001.html#T115">GLIB_001:115</a></i>;<br/><a NAME="E10:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">P</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">m</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">n</font> being  <a href="abian.html#V1">odd</a> <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E12:116_1_1_1_1_2_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E13:116_1_1_1_1_2_1"/>
<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E14:116_1_1_1_1_2_1"/><i><font color="Green">E68</font></i>: 
(  ex <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font> &amp; ( <font color="Maroon">P</font> is <a href="glib_001.html#V8">Cycle-like</a> implies ( (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 ) &amp; (  not <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 3 or  not <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font> ) ) ) )
 <b>by </b><i/>;<br/><a NAME="E15:116_1_1_1_1_2_1"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  )
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>consider </b><font color="Maroon">e</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:116_1_1_1_1_2_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">G</font>
 <b>by </b><i/>;<br/><a NAME="E18:116_1_1_1_1_2_1"/>
<font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> 2
 <b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E19:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E20:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><a NAME="E21:116_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> &amp; <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K25">the_Vertices_of</a> <font color="Maroon">H</font> )
 <b>by </b><i>, , <a class="ref" href="glib_001.html#T8">GLIB_001:8</a></i>;<br/><a NAME="E22:116_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">e</font> <a href="glib_000.html#R1">Joins</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">m</font>,<font color="Maroon">W</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>,<font color="Maroon">H</font>
 <b>by </b><i><a class="ref" href="chord.html#T19" target="_self">Th19</a>, <a class="ref" href="chord.html#T21" target="_self">Th21</a>, </i>;<br/><b>hence </b><a NAME="E23:116_1_1_1_1_2_1"/>
<font color="Maroon">W</font> is <a href="chord.html#V5" target="_self">chordal</a>
 <b>by </b><i>, , <a class="ref" href="glib_001.html#D31">GLIB_001:def 31</a>, , , </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:116_1_1_1_1_2"/>
<font color="Maroon">H</font> is <a href="chord.html#V6" target="_self">chordal</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:116_1_1"/>
<font color="Maroon">H</font> is <a href="chord.html#V6" target="_self">chordal</a>
 ;<br/>


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